Control charts for high-quality processes: MAX or CUMAX?

For attribute data with (very) small failure rates control charts were introduced which are based on subsequent groups of r failure times, for some r ≥ 1. Within this family, it was shown to be attractive to stop once the maximum of such a group is sufficiently small, because this choice allows a very satisfactory nonparametric adaptation. The question we address here is whether a cumulative approach offers even further improvement. Thus instead of fixed groups, we shall use the first sequence of r consecutive sufficiently small failure times to produce a signal. A further reason for considering this type of chart is the fact that it forms the nonparametric counterpart of the well-known sets method.